a real intertemporal model with investment · t1: current real taxes, t2: future real taxes are...

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Copyright © 2005 Pearson Education and Dr Yunus Aksoy. Slide 1

Test Thursday, 16th of February starting at 18:00 (ca. an hour), B34There will be a session (solving last year’s test) Tuesday 14th of February at 18:00


A Real Intertemporal Model with Investment

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So far

Work / leisure (labour supply) choice of the consumersIntertemporal consumption decisions of consumersLabour demand of firmsTotal factor productivity shocks/government expenditure shocksGovernment choice of tax timing


This week

A complete dynamic general equilibrium model without moneyThat is: ignore for the moment nominal issues

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Investment

investment : goods that are produced now for future use in the prod’n process (expenditure on plants/housing)

For the economy: Trade off between current consumption and future consumption

Instead of producing consumption goods engage in investment goods production thereby enhance productive capacity

For the firm: Trade off between higher profits today and higher future profits in the future (via increasing productive capacity)

Determinant: Real interest rates; opportunity cost of investment


Real interest rates

Higher interest rates implies higher opportunity cost of investment at the margin

A key determinant of monetary policy transmission (will be done in two weeks time!)

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What we will do

Look into the impact of exogenous shocks of capital stock (wars/earthquakes), fiscal policy, total factor productivity on equilibrium output, investment, consumption, the real interest rate, employment


Agents/ Markets

RepCon, (provides labour supply, consumes), RepFirm (demands labour, demands investment goods, and produces), Government (consumes)

Goods market (current/future)Labour market (current/future) clear

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Representative Consumer

Bring together work leisure choice and intertemporal consumption decision

Consumer works and consumes now and in the futureh units of time; allocates btw work and leisure in both periodsw1: current real wage, w2: future real wage are givenr: real interest rateT1: current real taxes, T2: future real taxes are givenDecision variables C1, C2 and l1, l2 and savings subject to budget and time constraints in the current and future periods


Consumers budget constraint

1 1 1 1 1

2 2 2 2 2

2 2 2 2 21 1 1 1 1

( )

( ) (1 )

( )( )1 1

p

p

Current periodC S w h l TFuture periodC w h l T r S

together

C w h l TC w h l Tr r

π

π

ππ

+ = − + −

= − + − + +

− + −+ = − + − +

+ +

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Four dimensional problem: choosing future and current consumption and leisure

Key optimality conditions1. MRSl1,C1 =w1

2. MRSl2,C2 =w2

3. MRSC1,C2 =1+r


Current Labour Supply

Increases in current real wage (assuming substitution effect dominates income effect)Increases in real interest rate

intertemporal substitution of leisure / labour supplyDecreases in lifetime wealth (π-T)

Current leisure and consumption are normal goods

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Income and Substitution Effects of an Increase in Total Factor Productivity


The Representative Consumer’s Current Labor Supply Curve

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Figure 7-2 An Increase in the Real Interest Rate Shifts the Current Labor Supply Curve to the Right


Figure 7-3 Effects of an Increase in Lifetime Wealth

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Current demand for consumption

decreases in real interest rates (if substitution effect dominates income effect) due to intertemporal substitution of consumption

Increases in lifetime wealth


Figure 7-4 The Representative Consumer’s Current Demand for Consumption Goods Increases with Income

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Figure 12 An Increase in the Real Interest Ratefor a Lender


Figure 7-5 An Increase in the Real Interest Rate from r1to r2 Shifts the Demand for Consumption Goods Down

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Figure 7-6 An Increase in Future Income Increases Lifetime Wealth for the Consumer, Shifting Up the Demand for Consumption Goods


The representative firm

Y1=z1F(K1,N1)

Y2=z2F(K2,N2)

Assume that one unit of consumption is needed to produce one unit of investmentFuture capital stock

K2=(1-d)K1+I

Note that by the end of second period capital stock is liquidated.

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Profits and Current Labour Demand

Current profits π1=Y1-w1N1-I1

Future profits π2=Y2-w2N2+(1-d)K2

Maximization problem:

V=π1+π2/(1+r)

Decision variables: N1 , N2, I

Firm hires up to the point where MPN=w


Figure 7-7 The Demand Curve for Current Labor Is the Representative Firm’s Marginal Product of Labor Schedule

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Figure 7-8 The Current Demand Curve for Labor Shifts Due to Changes in Current Total Factor Productivity z and in the Current Capital Stock K


Optimal Investment Decision

MC of investment should be equal to marginal benefit of investment MC(I) is engaging in investment by giving up one unit of net present value of profit (thus MC(I)=1)MB of investment is what one unit of extra investment yields in terms of net present value of profits

Two elementsAdditional unit of investment adds one unit in to future capital stock, future output increases; additional product is = future MPK Each unit of current investment implies an 1-d units of capital that will be liquidated at the end of the future period (remember there are only two periods!)

So with discounting2 1( )1KM P dM B I

r+ −

=+

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Optimality requires MC(I)=MB(I)

Firm invests until the net MPK is equal the real interest rate.Opportunity cost of investment is the real interest rate (i.e. return on alternative asset,bonds)

2KM P d r− =


Figure 7-9 Optimal Investment Schedule for the Representative Firm

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Optimal investment schedule shifts to the right

When future TFP increasesExpected future MPK higher, so increase investment

When the current capital stock is lowerFrom K2=(1-d)K1+I, a lower K1 means lower K2 for a given investmentLess initial capital will be left in the future after depreciationFuture MPK will be higher for each level of investment

Implication: variability of investment (unlike consumption)


Figure 7-10 The Optimal Investment Schedule Shifts to the Right if Current Capital Stock Decreases or Future Total Factor Productivity Is Expected to Increase

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Government

****2 2

1 1* *1 1G TG T

r r+ = +

+ +


Competitive Eq’m

Labour and goods market should clear both in the current and future periods!

(focus on the current period)

We need to construct the output supply and demand curves

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Figure 7-11 Determination of Equilibrium in the Labor Market Given the Real Interest Rate r


Figure 7-12 Construction of the Output Supply Curve

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What shifts Output Supply Curve to the RIGHT

Lifetime wealth decreases (say due to an increase in intertemporal government spending) income effect on labour supplyCurrent TFP increases more output for any level of input more labour demandCurrent capital stock increases more output


Figure 7-13 An Increase in Current or Future Government Spending Shifts the Ys Curve

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Figure 7-14 An Increase in Total Factor Productivity Shifts the Ys Curve


Figure 7-15 The Demand for Current Goods

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Figure 7-16 Construction of the Output Demand Curve


What shifts Output Demand Curve to the right ?

Increase in Lifetime wealth (via changes in taxes/government expenditures; fiscal stimulus)Increase in Future income; (consumption demand)Increase in Future TFP; (investment demand)Decrease in Current capital stock (via investment demand)

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Figure 7-17 The Output Demand Curve Shifts to the Right if Current Government Spending Increases


Figure 7-18 The Complete Real Intertemporal Model

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Experiments

A temporary increase in government expendituresA permanent increase in government expendituresDecrease in current capital stockIncrease in current TFPIncrease in future TFP


Figure 7-19 A Temporary Increase in Government Purchases

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Figure 7-20 A Permanent Increase in Government Purchases (PIH)


empirically

It seems that output effect is stronger when the fiscal stimulus is permanent (Aiyagari et al. (1992)

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Figure 7-21 Natural Log of Real Investment, 1929-1998


Figure 7-22 The Equilibrium Effects of a Decrease in the Current Capital Stock

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Figure 7-23 The Equilibrium Effects of an Increase in Current Total Factor Productivity


Figure 7-26 Percentage Deviations from Trend in Real Investment and Real GDP for the United States

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Figure 7-27 Percentage Deviations from Trend in Employment and Real GDP for the United States


Figure 7-28 Percentage Deviations from Trend in Real Consumption and Real GDP for the United States

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Figure 7-29 The Equilibrium Effects of an Increase in Future Total Factor Productivity (future MPL and MPK increase;

current investment goes up)

Copyright © Pearson Education, Inc. and Yunus Aksoy Slide 54

Summary of Comovements II

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In sum

InvestmentReal interest rates are key determinantsTemporary versus permanent shocksCurrent versus future shocks

a real intertemporal model with investment · t1: current real taxes, t2: future real taxes are...

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